Spin Hall effect in infinitely large and finite-size diffusive Rashba two-dimensional electron systems: A helicity-basis nonequilibrium Green's function approach

A nonequilibrium Green's function approach is employed to investigate the spin-Hall effect in diffusive two-dimensional electron systems with Rashba spin-orbit interaction. Considering a long-range electron-impurity scattering potential in the self-consistent Born approximation, we find that the spin-Hall effect arises from two distinct interband polarizations in helicity basis: a disorder-unrelated polarization directly induced by the electric field and a polarization mediated by electron-impurity scattering. The disorder-unrelated polarization is associated with all electron states below the Fermi surface and produces the original intrinsic spin-Hall current, while the disorder-mediated polarization emerges with contribution from the electron states near the Fermi surface and gives rise to an additional contribution to the spin-Hall current. Within the diffusive regime, the total spin-Hall conductivity vanishes in infinitely large samples, independently of temperature, of the spin-orbit coupling constant, of the impurity density, and of the specific form of the electron-impurity scattering potential. However, in a finite-size Rashba two-dimensional semiconductor, the spin-Hall conductivity no longer always vanishes. Depending on the sample size in the micrometer range, it can be positive, zero or negative with a maximum absolute value reaching as large as $e∕8\ensuremath{\pi}$ order of magnitude at low temperatures. As the sample size increases, the total spin-Hall conductivity oscillates with a decreasing amplitude. We also discuss the temperature dependence of the spin-Hall conductivity for different sample sizes.